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Given a singular modulus $j_0$ and a set of rational primes S, we study the problem of effectively determining the set of singular moduli j such that $j-j_0$ is an S-unit. For every $j_0 \neq 0$ , we provide an effective way of finding this set for infinitely many choices of S. The same is true if $j_0=0$ and we assume the Generalised Riemann Hypothesis. Certain numerical experiments will also lead to the formulation of a “uniformity conjecture” for singular S-units.

Using an analogue of Serre’s open image theorem for elliptic curves with complex multiplication, one can associate to each CM elliptic curve E E defined over a number field F F a natural number I ( E / F ) \mathcal {I}(E/F) which describes how big the image of the Galois representation associated to E E is. We show how one can compute I ( E / F ) \mathcal {I}(E/F) , using a closed formula that we obtain from the classical theory of complex multiplication.